# Density Ratio Estimation via Sampling along Generalized Geodesics on Statistical Manifolds

arXiv: https://arxiv.org/abs/2406.18806 · HTML: https://mlsanity.com/paper/2406.18806/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 59 references that are in the map.
- First posted on arXiv 27 Jun 2024.

## Builds on (most cited first, 8 of 8)

- [What is the Effect of Importance Weighting in Deep Learning?](https://mlsanity.com/paper/1812.03372/) (2018)
- [Telescoping Density-Ratio Estimation](https://mlsanity.com/paper/2006.12204/) (2020)
- [Density Ratio Estimation via Infinitesimal Classification](https://mlsanity.com/paper/2111.11010/) (2021)
- [Featurized Density Ratio Estimation](https://mlsanity.com/paper/2107.02212/) (2021)
- [A Short Survey on Importance Weighting for Machine Learning](https://mlsanity.com/paper/2403.10175/) (2024)
- [Generalization of Change-Point Detection in Time Series Data Based on Direct Density Ratio Estimation](https://mlsanity.com/paper/2001.06386/) (2020)
- [Double-Weighting for Covariate Shift Adaptation](https://mlsanity.com/paper/2305.08637/) (2023)
- [Information Geometrically Generalized Covariate Shift Adaptation](https://mlsanity.com/paper/2304.09387/) (2023)

## Built on by (most cited first, 8 of 8)

- [α-Flow: A Unified Framework for Continuous-State Discrete Flow Matching Models](https://mlsanity.com/paper/2504.10283/) (2025)
- [Density Ratio Estimation with Conditional Probability Paths](https://mlsanity.com/paper/2502.02300/) (2025)
- [Dequantified Diffusion-Schr"odinger Bridge for Density Ratio Estimation](https://mlsanity.com/paper/2505.05034/) (2025)
- [Diffusion Secant Alignment for Score-Based Density Ratio Estimation](https://mlsanity.com/paper/2509.04852/) (2025)
- [A Minimum Variance Path Principle for Accurate and Stable Score-Based Density Ratio Estimation](https://mlsanity.com/paper/2602.00834/) (2026)
- [Bounds on Lp errors in density ratio estimation via f-divergence loss functions](https://mlsanity.com/paper/2410.01516/) (2024)
- [Graph-Smoothed Bayesian Black-Box Shift Estimator and Its Information Geometry](https://mlsanity.com/paper/2505.16251/) (2025)
- [One-Step Score-Based Density Ratio Estimation](https://mlsanity.com/paper/2604.10672/) (2026)

## Abstract (arXiv)

The density ratio of two probability distributions is one of the fundamental tools in mathematical and computational statistics and machine learning, and it has a variety of known applications. Therefore, density ratio estimation from finite samples is a very important task, but it is known to be unstable when the distributions are distant from each other. One approach to address this problem is density ratio estimation using incremental mixtures of the two distributions. We geometrically reinterpret existing methods for density ratio estimation based on incremental mixtures. We show that these methods can be regarded as iterating on the Riemannian manifold along a particular curve between the two probability distributions. Making use of the geometry of the manifold, we propose to consider incremental density ratio estimation along generalized geodesics on this manifold. To achieve such a method requires Monte Carlo sampling along geodesics via transformations of the two distributions. We show how to implement an iterative algorithm to sample along these geodesics and show how changing the distances along the geodesic affect the variance and accuracy of the estimation of the density ratio. Our experiments demonstrate that the proposed approach outperforms the existing approaches using incremental mixtures that do not take the geometry of the